Question

Let W be the subspace of all diagonal matrices in M_{2,2}. Find a bais for W. Then give the dimension of W. If you need to enter a matrix as part of your answer , write each row as a vector.For example , write the matrix

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asked 2021-01-28
Let W be the subspace of all diagonal matrices in \(M_{2,2}\). Find a bais for W. Then give the dimension of W.
If you need to enter a matrix as part of your answer , write each row as a vector.For example , write the matrix

Answers (1)

2021-01-29
Step 1
Given that W is the subspace of all diagonal matrices in \(M_{2,2}\)
The objective is to a basis for W.
Step 2
Consider the given vector space W of all diagonal matrices in \(M_{2,2}\)
Therefore,
\(W=\left\{\begin{bmatrix}a & 0 \\0 & b \end{bmatrix}, \text{a and b can be any real number}\right\}\)
Let E be basis for W.
Therefore,
\(\begin{bmatrix}a & 0 \\0 & b \end{bmatrix}=a\begin{bmatrix}1 & 0 \\0 & 0 \end{bmatrix}+b\begin{bmatrix}0 & 0 \\0 & 1 \end{bmatrix}\)
Now, let \(\lambda_1\) and \(\lambda_2\) be any two scalars such that:
\(\lambda_1\begin{bmatrix}1 & 0 \\0 & 0 \end{bmatrix}+\lambda_2\begin{bmatrix}0 & 0 \\0 & 1 \end{bmatrix}=[]\)
\(\begin{bmatrix}\lambda_1 & 0 \\0 & 0 \end{bmatrix}+\begin{bmatrix}0 & 0 \\0 & \lambda_2 \end{bmatrix}=\begin{bmatrix}0 & 0 \\0 & 0 \end{bmatrix}\)
\(\begin{bmatrix}\lambda_1 & 0 \\0 & \lambda_2 \end{bmatrix}=\begin{bmatrix}0 & 0 \\0 & 0 \end{bmatrix}\)
Equating the elements:
\(\lambda_1=0 ,\lambda_2=0\)
Therefore, \(\begin{bmatrix}1 & 0 \\0 & 0 \end{bmatrix}\) and \(\begin{bmatrix}0 & 0 \\0 & 1 \end{bmatrix}\) are linear independent.
Hence, the basis of W is \(E=\left\{\begin{bmatrix}1 & 0 \\0 & 0 \end{bmatrix},\begin{bmatrix}0 & 0 \\0 & 1 \end{bmatrix}\right\}\) and dimension is 2.
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